### John Nachbar

### Washington University April 2, 2015

## The Envelope Theorem

^{1}

### 1 Introduction.

### The Envelope theorem is a corollary of the Karush-Kuhn-Tucker theorem (KKT) that characterizes changes in the value of the objective function in response to changes in the parameters in the problem. For example, in a standard cost mini- mization problem for a firm, the Envelope theorem characterizes changes in cost in response to changes in input prices or output quantities.

### 2 The Envelope Theorem with no binding constraints.

### Consider the following parameterized version of a differentiable MAX problem with no (binding) constraints,

### max

x### f (x, q)

### where q ∈ R

^{L}

### is a vector of parameters. The parametrized MIN problem is analo- gous. Let φ(q) give the (or at least, a) optimal x for a given q. The value function f

^{v}

### is defined by

### f

^{v}

### (q) = f (φ(q), q).

### The following is the Envelope theorem for this unconstrained problem. In theorem statement, D

x### f refers to the partial derivatives of f with respect to the x variables and D

_{q}

### f refers to the partial derivatives of f with respect to the q variables.

### Theorem 1. Fix a parametrized differentiable MAX or MIN problem. Let U ⊆ R

^{L}

### be open, let φ : U → R

^{N}

### be differentiable, and suppose that, for every q ∈ U , φ(q) is a solution to the MAX or MIN problem. Let f

^{v}

### : U → R be defined by f

^{v}

### (q) = f (φ(q), q). Finally, suppose that the standard first order condition, D

x### f (φ(q), q) = 0, holds for every q ∈ U . Then for any q

^{∗}

### ∈ U , letting x

^{∗}

### = φ(q

^{∗}

### ),

### Df

^{v}

### (q

^{∗}

### ) = D

q### f (x

^{∗}

### , q

^{∗}

### ).

1cbna. This work is licensed under the Creative Commons Attribution-NonCommercial- ShareAlike 4.0 License.

### Proof. By the Chain Rule

^{2}

### Df

^{v}

### (q

^{∗}

### ) = D

_{x}

### f (x

^{∗}

### , q

^{∗}

### )Dφ(q

^{∗}

### ) + D

_{q}

### f (x

^{∗}

### , q

^{∗}

### ).

### Since D

x### f (x

^{∗}

### , q

^{∗}

### ) = 0, the result follows.

### 3 An example with no binding constraints.

### Consider the cost minimization problem,

### min

_{a,b∈R}

### a + b s.t. √

### ab ≥ y a, b ≥ 0

### To interpret, a is capital, b is labor, and y is output. I assume (for simplicity) that the rental price of capital and the wage for labor both equal 1, so that total cost is a + b. If y > 0, any feasible a and b must be strictly positive, hence the non-negativity constraints do not bind at the solution. On the other hand, the production constraint √

### ab ≥ y binds (since cost is strictly increasing in both a and b, and since the production function is continuous), hence I can rewrite this constraint as

### b = y

^{2}

### a > 0.

### Substituting, let the (modified) objective function be f (a, y) = a + y

^{2}

### a .

### I can thus convert the above constrained minimization problem into an uncon- strained minimization problem

### min

_{a∈R}

### a +

^{y}

_{a}

^{2}

### .

### (With the qualification that I have to check that indeed a > 0 at any proposed solution.) In terms of notation, here a is playing the role of “x” and y is playing the role of “q”.

2Formally, define r : U → R^{N +L}by r(q) = (φ(q), q). Then f^{v}(q) = f (r(q)). By the Chain Rule,
Df^{v}(q^{∗}) = Df (x^{∗}, q^{∗})Dr(q^{∗})

=

Dxf (x^{∗}, q^{∗}) Dqf (x^{∗}, q^{∗})

Dφ(q^{∗})
I

= Dxf (x^{∗}, q^{∗})Dφ(q^{∗}) + Dqf (x^{∗}, q^{∗}).

as claimed. Many of the other Chain Rule applications are similar.

### At y = y

^{∗}

### , the first order condition D

_{a}

### f (a

^{∗}

### , y

^{∗}

### ) = 0 gives, 1 − y

^{∗2}

### a

^{∗2}

### = 0,

### hence y

^{∗}

### = y

^{∗}

### (which is strictly positive), hence φ(y) = a. For this particular application, call the value function C

^{`}

### (since the value function here is what is often called long-run cost). Then C

^{`}

### (y) = 2y.

### Verifying the Envelope theorem,

### DC

^{`}

### (y

^{∗}

### ) = D

_{y}

### f (a

^{∗}

### , y

^{∗}

### ), since

### D

_{y}

### f (a

^{∗}

### , y

^{∗}

### ) = 2y

^{∗}

### a

^{∗}

### = 2.

### Next, note that for a fixed (I won’t try to justify here in why a might be fixed), the value,

### a + y

^{2}

### a ,

### is the minimum cost of producing y (this is how I constructed the unconstrained problem in the first place). This can be interpreted as short-run cost, and accord- ingly I write,

### C

^{s}

### (a, y) = a + y

^{2}

### a .

### Short-run and long-run cost are equal when a = φ(y): C

^{`}

### (y) = C

^{s}

### (φ(y), y). By the Envelope theorem, then,

### DC

^{`}

### (y

^{∗}

### ) = D

_{y}

### C

^{s}

### (a

^{∗}

### , y

^{∗}

### ).

### For any given value of a

^{∗}

### , the graph of C

^{s}

### is strictly above that of C

^{`}

### (i.e., short- run cost is strictly higher than long-run cost) except at output level y

^{∗}

### , since at that output level, the level of the fixed factor a is optimal. Moreover, when y = y

^{∗}

### , so that φ(y

^{∗}

### ) = a

^{∗}

### , the two graphs are tangent; their slopes are equal. One says that the graph of C

^{`}

### is the (lower) envelope of the graph of C

^{s}

### , hence the name Envelope theorem. Graphing C

^{`}

### and a few of the C

^{s}

### will help you visualize what is going on.

### All of this generalizes. If C

^{s}

### is the short-run cost of producing a vector of outputs y given input prices and a fixed subvector of inputs a

^{∗}

### , and if a

^{∗}

### is optimal in the long-run when y = y

^{∗}

### , then

### DC

^{`}

### (y

^{∗}

### ) = D

_{y}

### C

^{s}

### (a

^{∗}

### , y

^{∗}

### ).

### 4 The General Envelope Theorem.

### Consider the following parameterized version of a differentiable MAX problem, max

x### f (x, q)

### s.t. g

_{1}

### (x, q) ≤ 0, .. . g

K### (x, q) ≤ 0.

### where q ∈ R

^{L}

### is a vector of parameters. For example, in a consumer maximization problem, the parameters are usually prices and income. The parameterized MIN problem is analogous. Again let φ(q) give the (or at least, a) optimal x for a given q and let the value function be defined by

### f

^{v}

### (q) = f (φ(q), q).

### Theorem 2 (Envelope Theorem). Fix a differentiable parameterized MAX or MIN problem. Let U ⊆ R

^{L}

### be open, let φ : U → R

^{N}

### be differentiable, and suppose that, for every q ∈ U , φ(q) is a solution to the MAX or MIN problem. Let f

^{v}

### : U → R be defined by f

^{v}

### (q) = f (φ(q), q). Let J (q) be the set of indices of constraints that are binding at φ(q). Finally, suppose that the conclusion of KKT holds for every q ∈ U . Fix any q

^{∗}

### ∈ U , let φ(q

^{∗}

### ) = x

^{∗}

### , let J

^{∗}

### = J (q

^{∗}

### ), and let λ

^{∗}

_{k}

### be the associated KKT multipliers. Then

### Df

^{v}

### (q

^{∗}

### ) = D

_{q}

### f (x

^{∗}

### , q

^{∗}

### ) − X

k∈J^{∗}

### λ

^{∗}

_{k}

### D

_{q}

### g

_{k}

### (x

^{∗}

### , q

^{∗}

### ).

### Proof. By the Chain Rule (see the proof of Theorem 1), Df

^{v}

### (q

^{∗}

### ) = D

x### f (x

^{∗}

### , q

^{∗}

### )Dφ(q

^{∗}

### ) + D

q### f (x

^{∗}

### , q

^{∗}

### ).

### By KKT,

### D

_{x}

### f (x

^{∗}

### , q

^{∗}

### ) = X

k∈J

### λ

^{∗}

_{k}

### D

_{x}

### g

_{k}

### (x

^{∗}

### , q

^{∗}

### ).

### Substituting,

### Df

^{v}

### (q

^{∗}

### ) = X

k∈J

### λ

^{∗}

_{k}

### D

_{x}

### g

_{k}

### (x

^{∗}

### , q

^{∗}

### )Dφ(q

^{∗}

### ) + D

_{q}

### f (x

^{∗}

### , q

^{∗}

### ).

### For any k ∈ J

^{∗}

### , let γ

_{k}

### (q) = g

_{k}

### (φ(q), q). For a MAC problem, since k is binding at q

^{∗}

### , γ

_{k}

### (q

^{∗}

### ) = 0 while for all other q ∈ U , γ

_{k}

### (q) ≤ 0 (since φ(q) must be feasible to be a solution). Therefore, q

^{∗}

### maximizes γ

k### on U . (For a MIN problem, the argument is almost the same but q

^{∗}

### minimizes γ

_{k}

### on U .) Therefore Dγ

_{k}

### (q

^{∗}

### ) = 0, hence, by the Chain Rule,

### 0 = Dγ

_{k}

### (q

^{∗}

### ) = D

x### g

_{k}

### (x

^{∗}

### , q

^{∗}

### )Dφ(q

^{∗}

### ) + D

q### g

_{k}

### (x

^{∗}

### , q

^{∗}

### ).

### Hence, D

_{x}

### g

_{k}

### (x

^{∗}

### , q

^{∗}

### )Dφ(q

^{∗}

### ) = −D

_{q}

### g

_{k}

### (x

^{∗}

### , q

^{∗}

### ), hence, since λ

^{∗}

_{k}

### ≥ 0, λ

^{∗}

_{k}

### D

_{x}

### g

_{k}

### (x

^{∗}

### , q

^{∗}

### )Dφ(q

^{∗}

### ) = −λ

^{∗}

_{k}

### D

_{q}

### g

_{k}

### (x

^{∗}

### , q

^{∗}

### ).

### Substituting this expression for λ

^{∗}

_{k}

### D

_{x}

### g

_{k}

### (x

^{∗}

### , q

^{∗}

### )Dφ(q

^{∗}

### ) into the above expression for Df

^{v}

### (q

^{∗}

### ) yields the result.

### 5 An example with constraints.

### Consider the problem

### max

x 13

### ln(x

1### ) +

^{2}

_{3}

### ln(x

2### ) s.t. p · x ≤ m,

### x ≥ 0

### Where x ∈ R

^{N}

### , p ∈ R

^{N}++

### and m ∈ R

++### . Think of this as a utility maximization problem with parameters being prices p 0 and income m > 0.

### Grinding through the Kuhn-Tucker calculation, at prices p

^{∗}

### and m

^{∗}

### the solution is,

### x

^{∗}

### =

### m

^{∗}

### /(3p

^{∗}

_{1}

### ) (2m

^{∗}

### )/(3p

^{∗}

_{2}

### )

### . Thus, at general (p, m), the solution function φ is given by,

### φ(p, m) =

### m/(3p

_{1}

### ) (2m)/(3p

2### )

### . Also

### λ

^{∗}

_{1}

### = 1 m

^{∗}

### and the other KKT multipliers (on the x ≥ 0 constraint) equal zero.

### The value function is then determined by f

^{v}

### (p, m) = f (φ(p, m)), hence f

^{v}

### (p, m) = 1

### 3 ln m 3p

_{1}

### + 2

### 3 ln 2m 3p

_{2}

### = ln(m) − 1

### 3 ln(p

_{1}

### ) − 2

### 3 ln(p

_{2}

### ) + 1 3 ln 1

### 3

### + 2

### 3 ln 2 3

### .

### Note that this makes some intuitive sense; in particular f

^{v}

### is increasing in m and decreasing in p

1### and p

2### .

### By direct calculation, at the point (p

^{∗}

### , m

^{∗}

### ),

### Df

^{v}

### (p

^{∗}

### , m

^{∗}

### ) =

###

###

###

∂f^{v}

∂p1

### (p

^{∗}

### , m

^{∗}

### )

∂f^{v}

∂p2

### (p

^{∗}

### , m

^{∗}

### )

∂f^{v}

∂m

### (p

^{∗}

### , m

^{∗}

### )

###

###

### =

###

###

### −1/(3p

^{∗}

_{1}

### )

### −2/(3p

^{∗}

_{2}

### ) 1/m

^{∗}

###

### .

### On the other hand, by the Envelope theorem, since the parameters p and m don’t appear in the objective function, and with the first constraint written in standard form as g

_{1}

### (x, p, m) = p · x − m ≤ 0:

### Df

^{v}

### (p

^{∗}

### , m

^{∗}

### ) =

###

###

### −λ

^{∗}

_{1}

### x

^{∗}

_{1}

### −λ

^{∗}

_{1}

### x

^{∗}

_{2}

### λ

^{∗}

_{1}

###

### .

### Substitute in the value of x

^{∗}

### and λ

^{∗}

### found above and you will confirm that these two expressions for Df

^{v}

### (p

^{∗}

### , m

^{∗}

### ) are indeed equal.

### 6 The constraint term and the interpretation of the λ

_{k}

### .

### Consider a MAX problem and focus on K parameters with the following special form. Parameter q

_{k}

### affects only constraint k and it does it additively:

### g

_{k}

### (x) − q

_{k}

### ≤ 0.

### Increasing q

k### weakens the constraint. Then, from the Envelope theorem, at q

^{∗}

_{k}

### = 0, D

_{q}

_{k}

### f

^{v}

### (0) = −λ

^{∗}

_{k}

### (−1) = λ

^{∗}

_{k}

### ,

### for any k ∈ J

^{∗}

### . That is, λ

^{∗}

_{k}

### is the marginal value of relaxing binding constraint k.

### The argument for MIN problems is almost identical.

### For k / ∈ J , set λ

^{∗}

_{k}